**Lemma** Suppose that $x\in \mathbb{R}$ with $x \geq 0$ and $x<1/n$ for every $n \in \mathbb{N}$. Then $x=0$. **Proof** Suppose $x \neq 0$. Then $x>0$. Using [[Archimedean Property of Real Numbers]], we can find $m \in \mathbb{N}$ such that $\frac{1}{m}<x$ contradicting our hypothesis so we must have $x=0$.